Fill a large pot with lightly salted water and bring to a boil. Stir in the linguine, and return to a boil. Cook the pasta uncovered, stirring occasionally, until cooked through, but still firm to the bite, about 11 minutes. Drain well in a colander; cover and keep warm.
Heat the olive oil over medium heat in a large pot with a cover that will hold a steamer basket. Add the mushrooms; stir and cook for 3 minutes. Pour in the wine and bring to a boil; add the shrimp.
Put the clams and asparagus into the steamer basket and place the basket into the pot over the shrimp and mushrooms. Cover and steam for 2 minutes. Lift the steamer basket and add the scallops to the shrimp and mushrooms. Cover and continue to steam until clam shells open and seafood is cooked, about 4 minutes.
Tip the clams and asparagus into the pot and combine with the shrimp and scallops. To serve, ladle seafood over the cooked linguine and sprinkle with grated Parmesan cheese.